Theorem ipasslem1 25408

Description: Lemma for ipassi 25418. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	|- X = ( BaseSet ` U )
ip1i.2	|- G = ( +v ` U )
ip1i.4	|- S = ( .sOLD ` U )
ip1i.7	|- P = ( .iOLD ` U )
ip1i.9	|- U e. CPreHil OLD
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem1	|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Proof of Theorem ipasslem1

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 10794	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
2		ax-1cn 9539	. . . . . . . . . . . 12 |- 1 e. CC
3		ip1i.9	. . . . . . . . . . . . . 14 |- U e. CPreHil OLD
4	3	phnvi 25393	. . . . . . . . . . . . 13 |- U e. NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 |- X = ( BaseSet ` U )
6		ip1i.2	. . . . . . . . . . . . . 14 |- G = ( +v ` U )
7		ip1i.4	. . . . . . . . . . . . . 14 |- S = ( .sOLD ` U )
8	5, 6, 7	nvdir 25188	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( k e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
9	4, 8	mpan 670	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
10	2, 9	mp3an2 1307	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
11	1, 10	sylan 471	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
12	5, 7	nvsid 25184	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
13	4, 12	mpan 670	. . . . . . . . . . . 12 |- ( A e. X -> ( 1 S A ) = A )
14	13	adantl 466	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 S A ) = A )
15	14	oveq2d 6291	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k S A ) G ( 1 S A ) ) = ( ( k S A ) G A ) )
16	11, 15	eqtrd 2501	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G A ) )
17	16	oveq1d 6290	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) G A ) P B ) )
18		ipasslem1.b	. . . . . . . . . . . . 13 |- B e. X
19		ip1i.7	. . . . . . . . . . . . . 14 |- P = ( .iOLD ` U )
20	5, 19	dipcl 25287	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
21	4, 18, 20	mp3an13 1310	. . . . . . . . . . . 12 |- ( A e. X -> ( A P B ) e. CC )
22	21	mulid2d 9603	. . . . . . . . . . 11 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
23	22	adantl 466	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 x. ( A P B ) ) = ( A P B ) )
24	23	oveq2d 6291	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
25	5, 7	nvscl 25183	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ k e. CC /\ A e. X ) -> ( k S A ) e. X )
26	4, 25	mp3an1 1306	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( k S A ) e. X )
27	1, 26	sylan 471	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( k S A ) e. X )
28	5, 6, 7, 19, 3	ipdiri 25407	. . . . . . . . . . 11 |- ( ( ( k S A ) e. X /\ A e. X /\ B e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
29	18, 28	mp3an3 1308	. . . . . . . . . 10 |- ( ( ( k S A ) e. X /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
30	27, 29	sylancom 667	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
31	24, 30	eqtr4d 2504	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) G A ) P B ) )
32	17, 31	eqtr4d 2504	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) )
33		oveq1 6282	. . . . . . 7 |- ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
34	32, 33	sylan9eq 2521	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
35		adddir 9576	. . . . . . . . 9 |- ( ( k e. CC /\ 1 e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
36	2, 35	mp3an2 1307	. . . . . . . 8 |- ( ( k e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
37	1, 21, 36	syl2an 477	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
38	37	adantr 465	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
39	34, 38	eqtr4d 2504	. . . . 5 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) )
40	39	exp31 604	. . . 4 |- ( k e. NN0 -> ( A e. X -> ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
41	40	a2d 26	. . 3 |- ( k e. NN0 -> ( ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
42		eqid 2460	. . . . . 6 |- ( 0vec ` U ) = ( 0vec ` U )
43	5, 42, 19	dip0l 25293	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
44	4, 18, 43	mp2an 672	. . . 4 |- ( ( 0vec ` U ) P B ) = 0
45	5, 7, 42	nv0 25194	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( 0vec ` U ) )
46	4, 45	mpan 670	. . . . 5 |- ( A e. X -> ( 0 S A ) = ( 0vec ` U ) )
47	46	oveq1d 6290	. . . 4 |- ( A e. X -> ( ( 0 S A ) P B ) = ( ( 0vec ` U ) P B ) )
48	21	mul02d 9766	. . . 4 |- ( A e. X -> ( 0 x. ( A P B ) ) = 0 )
49	44, 47, 48	3eqtr4a 2527	. . 3 |- ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) )
50		oveq1 6282	. . . . . 6 |- ( j = 0 -> ( j S A ) = ( 0 S A ) )
51	50	oveq1d 6290	. . . . 5 |- ( j = 0 -> ( ( j S A ) P B ) = ( ( 0 S A ) P B ) )
52		oveq1 6282	. . . . 5 |- ( j = 0 -> ( j x. ( A P B ) ) = ( 0 x. ( A P B ) ) )
53	51, 52	eqeq12d 2482	. . . 4 |- ( j = 0 -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) )
54	53	imbi2d 316	. . 3 |- ( j = 0 -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) ) )
55		oveq1 6282	. . . . . 6 |- ( j = k -> ( j S A ) = ( k S A ) )
56	55	oveq1d 6290	. . . . 5 |- ( j = k -> ( ( j S A ) P B ) = ( ( k S A ) P B ) )
57		oveq1 6282	. . . . 5 |- ( j = k -> ( j x. ( A P B ) ) = ( k x. ( A P B ) ) )
58	56, 57	eqeq12d 2482	. . . 4 |- ( j = k -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) )
59	58	imbi2d 316	. . 3 |- ( j = k -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) ) )
60		oveq1 6282	. . . . . 6 |- ( j = ( k + 1 ) -> ( j S A ) = ( ( k + 1 ) S A ) )
61	60	oveq1d 6290	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j S A ) P B ) = ( ( ( k + 1 ) S A ) P B ) )
62		oveq1 6282	. . . . 5 |- ( j = ( k + 1 ) -> ( j x. ( A P B ) ) = ( ( k + 1 ) x. ( A P B ) ) )
63	61, 62	eqeq12d 2482	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) )
64	63	imbi2d 316	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
65		oveq1 6282	. . . . . 6 |- ( j = N -> ( j S A ) = ( N S A ) )
66	65	oveq1d 6290	. . . . 5 |- ( j = N -> ( ( j S A ) P B ) = ( ( N S A ) P B ) )
67		oveq1 6282	. . . . 5 |- ( j = N -> ( j x. ( A P B ) ) = ( N x. ( A P B ) ) )
68	66, 67	eqeq12d 2482	. . . 4 |- ( j = N -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
69	68	imbi2d 316	. . 3 |- ( j = N -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) ) )
70	41, 49, 54, 59, 64, 69	nn0indALT 10947	. 2 |- ( N e. NN0 -> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
71	70	imp 429	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   NN0cn0 10784   NrmCVeccnv 25139   +vcpv 25140   BaseSetcba 25141   .sOLDcns 25142   0veccn0v 25143   .iOLDcdip 25272   CPreHil OLDccphlo 25389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-grpo 24855  df-gid 24856  df-ginv 24857  df-ablo 24946  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-nmcv 25155  df-dip 25273  df-ph 25390

This theorem is referenced by:  ipasslem2  25409  ipasslem3  25410  ipasslem4  25411